Graphic sequences with a realization containing a complete multipartite subgraph

A nonincreasing sequence of nonnegative integers π=(d1,d2,…,dn)π=(d1,d2,…,dn) is graphic   if there is a (simple) graph GG of order nn having degree sequence ππ. In this case, GG is said to realize  ππ. For a given graph HH, a graphic sequence ππ is potentially  HH-graphic   if there is some realization of ππ containing HH as a (weak) subgraph. Let σ(π)σ(π) denote the sum of the terms of ππ. For a graph HH and n∈Z+n∈Z+, σ(H,n)σ(H,n) is defined as the smallest even integer mm so that every nn-term graphic sequence ππ with σ(π)≥mσ(π)≥m is potentially HH-graphic. Let Kst denote the complete tt partite graph such that each partite set has exactly ss vertices. We show that σ(Kst,n)=σ(K(t−2)s+Ks,s,n) and obtain the exact value of σ(Kj+Ks,s,n)σ(Kj+Ks,s,n) for nn sufficiently large. Consequently, we obtain the exact value of σ(Kst,n) for nn sufficiently large.